1. Field of the Invention
This disclosure relates to analyzers for electrical circuits. In particular, this disclosure relates to an apparatus and method for analyzing linear and linearized circuits and networks to approximate a transfer function in terms of dominant poles, zeroes, and residues.
2. Description of the Related Art
Circuit analysis using probe apparatus and circuit simulators is employed to determine the behavior and response of the interconnected elements of a circuit. As the complexity or degree of interconnectedness of a network of elements and/or the number of elements increases (i.e. becomes "larger") the accurate prediction and full accounting of interconnect effects of various elements, especially if transmission lines, ground planes, antennas, and the like are included, often requires increased computational complexity in circuit analyzers. For example, SPICE-like simulators based on integration of non-linear ordinary differential equations are useful in analyzing nonlinear circuits, but are relatively inefficient for highly interconnected (or "large") circuits or networks.
Techniques are known in the art for large linear circuit analysis. One technique in particular is the Asymptotic Waveform Evaluation (AWE) algorithm described in L. T. Pillage and R. A. Rohrer, Asymptotic Waveform Evaluation for Timing Analysis, IEEE TRANS. COMPUTER-AIDED DESIGN, Vol. CAD-9, pp. 352-366, April 1990; in X. Huang, Pade Approximation of Linear(ized) Circuit Responses, Ph.D. Dissertation, Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, Pa., November, 1990; and in V. Raghavan, R. A. Rohrer, T. L. Pillage et al., AWE-Inspired, PROC. OF THE IEEE 1993 CUSTOM INTEGRATED CIRCUITS CONF., May 1993.
AWE is based on approximating the Laplace domain transfer function H(s) of a linear network by a reduced order model using Pade Approximation by computing the leading 2 q moments me, m.sub.0, m.sub.1 . . . , m.sub.2q-1 of H(s).
Using circuit equation formulation methods such as modified nodal analysis, sparse tableau, etc. such as described in J. Vlach and K. Singhal, COMPUTER METHODS FOR CIRCUIT ANALYSIS AND DESIGN, Van Nostrand Reinhold, New York, N.Y., 1983; a lumped, linear, time-invariant circuit may be described by the following system of first order differential equations: EQU Cx=Gx+bu (1) EQU y=l.sup.T x+du
where the vector x represents the circuit variables; matrix G represents the contribution of memoryless elements, such as resistors; C represents the contribution from memory elements, such as capacitors and inductors; 1 is an output selection vector for choosing the variables of interest in the circuit for analysis; y is the output of interest; and the terms bu and du represent excitations from independent sources. Hereinafter, variables are scalars unless identified as vectors or matrices, including matrices of vectors.
In determining the impulse response of the linear circuit with zero initial conditions (which is used to determine response behavior to any excitation) the Laplace transformation of the system in Eq. (1) above, assuming zero initial conditions, becomes EQU sCX=-GX+bU (2) EQU Y=l.sup.T X
where X, U and Y denote the Laplace transform of x, u and y, respectively.
The Laplace domain impulse response, or transfer function, H(s) is defined as ##EQU1##
Hereinafter, the notation ":=" indicates assignment, definition, or setting a variable accordingly, and ( ).sup.-1 indicates the inverse operator for scalars and matrices, accordingly.
Let s.sub.0 .epsilon. be an arbitrary but fixed expansion point such that the matrix G+s.sub.0 C is non-singular, and using the change of variables s=s.sub.0 +.sigma., set EQU A:=-(G+S.sub.0 C).sup.- C (4) EQU r:=(G+s.sub.0 C).sup.-1 b
Matrix A is hereinafter called the characteristic matrix or circuit characteristic matrix of the circuit, determined from the circuit matrices G and C of the characteristics of the circuit or network.
Eq. (3) may be rewritten as follows: ##EQU2##
where I is the identity matrix.
To determine the impulse response H(s.sub.0 +.sigma.) from the characteristic matrix A, all of the eigenvalues and eigenvectors of A are to be determined. However, the numerical computation of all eigenvalues and eigenvectors of the matrix A becomes prohibitively expensive computationally for larger sizes of matrix A, so a Pade approximation to the network impulse response H(s.sub.0 +.sigma.) is used.
For each pair of integers p, q.gtoreq.0, the Pade approximation of type p/q to H(s.sub.0 +.sigma.) is defined as the rational function ##EQU3##
whose Taylor series about .sigma.=0 (i.e. the Maclaurin expansion) agrees with the Taylor series of H(s.sub.0 +.sigma.) in at least the first (p+q+1) terms.
The coefficients of the Pade approximation in Eq. (6) are uniquely determined by the first (p+q+1) Taylor coefficients of the impulse response, and the roots of the denominator and numerator polynomial in Eq. (6) represent approximation to the poles and zeros of the network, respectively.
In the context of impulse response approximations, one may choose p=q-1 so that the Pade approximation is of the same form as the original impulse response. Define H.sub.q :=H.sub.q-1,q. Using H.sub.q as the q.sup.th Pade approximant to the impulse response H, and by using a partial fraction decomposition, H.sub.q may be expressed in the form ##EQU4##
In AWE, the coefficients of the denominator polynomial of Eq. (6) of H.sub.q are obtained by solving linear systems of order q, as described E. Chiprout and M. S. Nakhla, ASYMPTOTIC WAVEFORM EVALUATION AND MOMENT MATCHING FOR INTERCONNECT ANALYSIS, Kluwer Academic Publishers, Norwell, Mass. 1994. However, as q increases, the computations of AWE to solve the linear system of order q described above is susceptible to extremely ill-conditioned numerical computations, especially in the case when the required order or degree of approximation is "large". A number of strategies such as scaling, frequency shifting, and complex frequency hopping have been proposed to remedy AWE, as described in X. Huang and in E. Chiprout et al., cited above. Such strategies have met with some success.
It is preferable to analyze large linear circuits or networks using a numerically stable method to determine the Pade approximation of the impulse response with little numerical degradation, as well as to provide an acceptable computational cost per order of approximation. It is also advantageous to obtain a qualitative measure of the accuracy of determined poles.